無限に続くルート問題

\begin{align*} \sqrt{2\sqrt{4\sqrt{8\sqrt{16\cdots}}}} & =\sqrt{2\sqrt{2^{2}\sqrt{2^{3}\sqrt{2^{4}\cdots}}}}\\ & =2^{\frac{1}{2}}2^{\frac{2}{4}}2^{\frac{3}{8}}2^{\frac{4}{16}}\cdots\\ & =\prod_{k=1}^{\infty}2^{\frac{k}{2^{k}}}\\ & =\exp\left(\log2\cdot\sum_{k=1}^{\infty}\frac{k}{2^{k}}\right)\\ & =\exp\left(-\log2\cdot\left[a\frac{d}{da}\sum_{k=1}^{\infty}\frac{1}{a^{k}}\right]_{a=2}\right)\\ & =\exp\left(-\log2\cdot\left[a\frac{d}{da}\frac{1}{a-1}\right]_{a=2}\right)\\ & =\exp\left(-\log2\cdot\left[-a\frac{1}{\left(a-1\right)^{2}}\right]_{a=2}\right)\\ & =\exp\left(2\log2\right)\\ & =4 \end{align*}